1. Field of the Invention
The present invention is directed to the field of dynamic system control. In particular, the present invention is directed to the field of improved open loop control of uncertain dynamical systems.
2. Description of the Related Technology
In control of dynamical systems, feedback is generally utilized to achieve a specified performance despite a range of system (plant) uncertainty. Such uncertainty may be the result of physical changes in the characteristics of the system under control that occur over time, for example due to age or use, or that occur due to changes in the environment, such as the change in density of the atmosphere as a function of altitude, season, latitude, longitude, and local time of day. In addition, system uncertainty may arise in the control system design due to the operator's lack of knowledge or ignorance of the true values of the system parameters. For example in the control of the angular rotation of a rigid body, an important system parameter is the inertia of the body about the rotational axis. In general, an estimate of the inertia is developed based on measurements of certain characteristics of the body such as its mass distribution and dimensions. However, since these parameters are often difficult to measure with precision, the exact value of the inertia is difficult to predict. As a consequence, it is difficult in practice to achieve a desired input-output relation due to imprecise knowledge of the system. The role of feedback is therefore to obtain the desired input-output relation within desired tolerances despite the system uncertainty, which may additionally include the effects of disturbances acting on or within the system.
In a feedback dynamic control system, sensors are used to measure the system variables or states. The sensor data is processed and fed back into the input of the system to force the system to produce an acceptable output despite uncertainty. In this setting, the system accuracy can be no greater than the accuracy of the sensors used to provide the feedback. As a consequence, the performance of the feedback system will become degraded when sensors fail. In the situation where the sensors fail completely, that is to say the sensors can no longer produce an output that allows the system states to be estimated, the feedback mechanism will also fail and it will become impossible to control the system. In this situation, it may still be possible to achieve a specified performance tolerance if a suitable open-loop control (one that does not rely on sensor data for implementation) can be designed and applied to the system.
In the past, feedback has been utilized to achieve the necessary sensitivity reduction by using sensors to measure the system state so that the system state may be compared with the desired state. As a consequence of this comparison, an error signal may be computed and subsequently filtered by a feedback law in order to generate a control input to drive the plant. By measuring the system states, any deviations caused by parametric uncertainties or other disturbances are automatically accommodated by the resulting changes in the error signal. In the event that some or all of the system sensors fail, it becomes impossible to correctly compute the error signal and as a consequence an incorrect control signal will be output by the feedback law. In general, this will cause the response of the dynamical system under control to drift outside of the desired tolerance. In some cases, the control system can become unstable causing the destruction of the system.
An approach for operating a dynamical system in the absence of, or degradation of feedback sensors, is to implement an open-loop control system by designing a suitable control history a priori and using it to drive the system directly. Designing an open-loop control history to correctly manage uncertainty and thereby achieve the desired tolerances for the system response is a considerable challenge and consequently very little has been done to address this issue.
One representative approach employs the concept of a system sensitivity function, obtained by linearizing the system dynamics that attempts to capture the effects of uncertainty in the neighborhood of the system response obtained under nominal conditions. Open-loop control is implemented by introducing a constraint that requires the sensitivity states to be forced to zero at or near the final time. One difficulty with this approach is that the construction of the sensitivity function is generally based on the linearization of a dynamical model of the system. As a result, uncertainties causing significant drifts may be difficult or impossible to accommodate. This can lead to failure of the open-loop control to achieve the desired objective(s).